## The Annals of Applied Probability

### The Widom–Rowlinson model under spin flip: Immediate loss and sharp recovery of quasilocality

#### Abstract

We consider the continuum Widom–Rowlinson model under independent spin-flip dynamics and investigate whether and when the time-evolved point process has an (almost) quasilocal specification (Gibbs-property of the time-evolved measure). Our study provides a first analysis of a Gibbs–non-Gibbs transition for point particles in Euclidean space. We find a picture of loss and recovery, in which even more regularity is lost faster than it is for time-evolved spin models on lattices.

We show immediate loss of quasilocality in the percolation regime, with full measure of discontinuity points for any specification. For the color-asymmetric percolating model, there is a transition from this non-almost-sure quasilocal regime back to an everywhere Gibbsian regime. At the sharp reentrance time $t_{G}>0$, the model is a.s. quasilocal. For the color-symmetric model, there is no reentrance. On the constructive side, for all $t>t_{G}$, we provide everywhere quasilocal specifications for the time-evolved measures and give precise exponential estimates on the influence of boundary condition.

#### Article information

Source
Ann. Appl. Probab., Volume 27, Number 6 (2017), 3845-3892.

Dates
Revised: March 2017
First available in Project Euclid: 15 December 2017

https://projecteuclid.org/euclid.aoap/1513328715

Digital Object Identifier
doi:10.1214/17-AAP1298

Mathematical Reviews number (MathSciNet)
MR3737939

Zentralblatt MATH identifier
06848280

#### Citation

Jahnel, Benedikt; Külske, Christof. The Widom–Rowlinson model under spin flip: Immediate loss and sharp recovery of quasilocality. Ann. Appl. Probab. 27 (2017), no. 6, 3845--3892. doi:10.1214/17-AAP1298. https://projecteuclid.org/euclid.aoap/1513328715

#### References

• [1] Bollobás, B. and Riordan, O. (2006). Percolation. Cambridge Univ. Press, New York.
• [2] Bricmont, J., Kuroda, K. and Lebowitz, J. L. (1984). The structure of Gibbs states and phase coexistence for nonsymmetric continuum Widom–Rowlinson models. Z. Wahrsch. Verw. Gebiete 67 121–138.
• [3] Cassandro, M. and Da Fano, A. (1974). Rigorous properties of a continuous system in the high activity region. Comm. Math. Phys. 36 277–286.
• [4] Chayes, J. T., Chayes, L. and Kotecký, R. (1995). The analysis of the Widom–Rowlinson model by stochastic geometric methods. Comm. Math. Phys. 172 551–569.
• [5] Chiu, S. N., Stoyan, D., Kendall, W. S. and Mecke, J. (2013). Stochastic Geometry and Its Applications, 3rd ed. Wiley, Chichester.
• [6] Conache, D., Daletskii, A., Kondratiev, Y. and Pasurek, T. (2015). Gibbs measures on marked configuration spaces: Existence and uniqueness. Available at arXiv:1503.06349.
• [7] den Hollander, F., Redig, F. and van Zuijlen, W. (2015). Gibbs–non-Gibbs dynamical transitions for mean-field interacting Brownian motions. Stochastic Process. Appl. 125 371–400.
• [8] Dereudre, D. (2016). Variational principle for Gibbs point processes with finite range interaction. Electron. Commun. Probab. 21 Paper No. 10, 11.
• [9] Dereudre, D., Drouilhet, R. and Georgii, H.-O. (2012). Existence of Gibbsian point processes with geometry-dependent interactions. Probab. Theory Related Fields 153 643–670.
• [10] De Roeck, W., Maes, C., Netočný, K. and Schütz, M. (2015). Locality and nonlocality of classical restrictions of quantum spin systems with applications to quantum large deviations and entanglement. J. Math. Phys. 56 023301, 30.
• [11] Ermolaev, V. and Külske, C. (2010). Low-temperature dynamics of the Curie–Weiss model: Periodic orbits, multiple histories, and loss of Gibbsianness. J. Stat. Phys. 141 727–756.
• [12] Fernández, R., den Hollander, F. and Martínez, J. (2014). Variational description of Gibbs–non-Gibbs dynamical transitions for spin-flip systems with a Kac-type interaction. J. Stat. Phys. 156 203–220.
• [13] Georgii, H.-O. (2011). Gibbs Measures and Phase Transitions, 2nd ed. De Gruyter Studies in Mathematics 9. de Gruyter, Berlin.
• [14] Georgii, H.-O. and Häggström, O. (1996). Phase transition in continuum Potts models. Comm. Math. Phys. 181 507–528.
• [15] Georgii, H.-O., Häggström, O. and Maes, C. (2001). The random geometry of equilibrium phases. In Phase Transitions and Critical Phenomena, Vol. 18. Phase Transit. Crit. Phenom. 18 1–142. Academic Press, San Diego, CA.
• [16] Georgii, H.-O., Schreiber, T. and Thäle, C. (2015). Branching random tessellations with interaction: A thermodynamic view. Ann. Probab. 43 1892–1943.
• [17] Higuchi, Y. and Takei, M. (2004). Some results on the phase structure of the two-dimensional Widom–Rowlinson model. Osaka J. Math. 41 237–255.
• [18] Hirsch, C., Jahnel, B., Patterson, R. I. A. and Keeler, P. (2016). Traffic flow densities in large transport networks. Available at arXiv:1602.01009.
• [19] Hug, D., Last, G. and Schulte, M. (2016). Second-order properties and central limit theorems for geometric functionals of Boolean models. Ann. Appl. Probab. 26 73–135.
• [20] Jansen, S. (2016). Continuum percolation for Gibbsian point processes with attractive interactions. Electron. J. Probab. 21 Paper No. 47, 22.
• [21] Jansen, S., König, W. and Metzger, B. (2015). Large deviations for cluster size distributions in a continuous classical many-body system. Ann. Appl. Probab. 25 930–973.
• [22] Janson, S. (1984). Bounds on the distributions of extremal values of a scanning process. Stochastic Process. Appl. 18 313–328.
• [23] Kondratiev, Y., Pasurek, T. and Röckner, M. (2012). Gibbs measures of continuous systems: An analytic approach. Rev. Math. Phys. 24 1250026, 54.
• [24] Külske, C., Le Ny, A. and Redig, F. (2004). Relative entropy and variational properties of generalized Gibbsian measures. Ann. Probab. 32 1691–1726.
• [25] Külske, C. and Opoku, A. A. (2008). The posterior metric and the goodness of Gibbsianness for transforms of Gibbs measures. Electron. J. Probab. 13 1307–1344.
• [26] Külske, C. and Redig, F. (2006). Loss without recovery of Gibbsianness during diffusion of continuous spins. Probab. Theory Related Fields 135 428–456.
• [27] Kutoviy, O. V. and Rebenko, A. L. (2004). Existence of Gibbs state for continuous gas with many-body interaction. J. Math. Phys. 45 1593–1605.
• [28] Last, G., Penrose, M. D., Schulte, M. and Thäle, C. (2014). Moments and central limit theorems for some multivariate Poisson functionals. Adv. in Appl. Probab. 46 348–364.
• [29] Lebowitz, J. L., Mazel, A. and Presutti, E. (1999). Liquid-vapor phase transitions for systems with finite-range interactions. J. Stat. Phys. 94 955–1025.
• [30] Le Ny, A. and Redig, F. (2002). Short time conservation of Gibbsianness under local stochastic evolutions. J. Stat. Phys. 109 1073–1090.
• [31] Peccati, G. and Reitzner, M., eds. (2016). Malliavin calculus, Wiener–Itô chaos expansions and stochastic geometry. In Stochastic Analysis for Poisson Point Processes. Bocconi & Springer Series 7. Bocconi Univ. Press.
• [32] Penrose, M. D. and Pisztora, A. (1996). Large deviations for discrete and continuous percolation. Adv. in Appl. Probab. 28 29–52.
• [33] Rœlly, S. and Ruszel, W. M. (2014). Propagation of Gibbsianness for infinite-dimensional diffusions with space–time interaction. Markov Process. Related Fields 20 653–674.
• [34] Ruelle, D. (1970). Superstable interactions in classical statistical mechanics. Comm. Math. Phys. 18 127–159.
• [35] Ruelle, D. (1971). Existence of a phase transition in a continuous classical system. Phys. Rev. Lett. 27 1040–1041.
• [36] Ruelle, D. (1999). Statistical Mechanics: Rigorous Results. World Scientific, River Edge, NJ.
• [37] van Enter, A. C. D., Ermolaev, V. N., Iacobelli, G. and Külske, C. (2012). Gibbs–non-Gibbs properties for evolving Ising models on trees. Ann. Inst. Henri Poincaré Probab. Stat. 48 774–791.
• [38] van Enter, A. C. D., Fernández, R., den Hollander, F. and Redig, F. (2002). Possible loss and recovery of Gibbsianness during the stochastic evolution of Gibbs measures. Comm. Math. Phys. 226 101–130.
• [39] van Enter, A. C. D., Fernández, R., den Hollander, F. and Redig, F. (2010). A large-deviation view on dynamical Gibbs–non-Gibbs transitions. Mosc. Math. J. 10 687–711, 838.
• [40] van Enter, A. C. D., Fernández, R. and Sokal, A. D. (1993). Regularity properties and pathologies of position-space renormalization-group transformations: Scope and limitations of Gibbsian theory. J. Stat. Phys. 72 879–1167.
• [41] van Enter, A. C. D., Külske, C., Opoku, A. A. and Ruszel, W. M. (2010). Gibbs–non-Gibbs properties for $n$-vector lattice and mean-field models. Braz. J. Probab. Stat. 24 226–255.
• [42] van Enter, A. C. D. and Le Ny, A. (2017). Decimation of the Dyson–Ising ferromagnet. Stochastic Process. Appl. Available at https://doi.org/10.1016/j.spa.2017.03.007.
• [43] Widom, B. and Rowlinson, J. S. (1970). New model for the study of liquid–vapor phase transitions. J. Chem. Phys. 52 1670–1684.